Question

In the adjoining figure, XY = XZ . YQ and ZP are the bisectors of
`\angle` XYZ and
`\angle` XZY respectively. Prove that YQ = ZP.

~Thanks in advance ! ♡

Answer 1

this is your answer. look at this once.

Answer 2

See Below.

Explanation:

Statements: Reasons:

`1)\, XY=XZ` Given

`2) \text{ $ m\angle Y= m\angle Z$}` Isosceles Triangle Theorem

`\displaystyle 3) \text{ $m\angle Y=m\angle XYQ + \angle QYZ$}` Angle Addition

`\displaystyle 4)\text{ $YQ$ bisects $\angle XYZ$}` Given

`5) \text{ $m\angle XYQ=m\angle QYZ$}` Definition of Bisector

`\displaystyle 6)\text{ $m\angle Y=2m\angle QYZ$}` Substitution

`7)\text{ $m\angle Z=m\angle XZP+m\angle PZY$}` Angle Addition

`8)\text{ $ZP$ bisects $\angle XZY$}` Given

`\displaystyle 9) \text{ $m\angle XZP=m\angle PZY$ }` Definition of Bisector

`\displaystyle 10) \text{ $ m\angle Z = 2m\angle PZY $}` Substitution

`11)\text{ } 2m\angle QYZ=2m\angle PZY` Substitution

`12)\text{ }m\angle QYZ=m\angle PZY` Division Property of Equality

`13)\text{ } YZ=YZ` Reflexive Property

`14)\text{ } \Delta YZP\cong\Delta ZYQ` Angle-Side-Angle Congruence*

`15)\text{ } YQ=ZP` CPCTC

*For clarification:

∠Y = ∠Z

YZ = YZ (or ZY)

∠PZY = ∠QYZ

So, Angle-Side-Angle Congruence:

ΔYZP is congruent to ΔZYQ